c 2006 Tech Science Press
Copyright CMES, vol.14, no.2, pp.91-100, 2006
A lattice-based cell model for calculating thermal capacity and expansion of single
wall carbon nanotubes
Xianwu Ling1 and S.N. Atluri
Abstract: In this paper, a lattice-based cell model is
proposed for single wall carbon nanotubes (SWNTs).
The finite temperature effect is accounted for via the local harmonic approach. The equilibrium SWNT configurations are obtained by minimizing the Helmholtz free
energy with respect to seven primary coordinate variables that are subjected to a chirality constraint. The
calculated specific heats agree well with the experimental data, and at low temperature depend on the tube radii
with small tubes having much lower values. Our calculated coefficients of thermal expansion (CTEs) are universally positive for all the radial, axial and circumferential directions, and increase with increasing temperature.
The armchair tubes see very large circumferential CTEs,
while the zigzag tubes see very large axial CTEs. The
tube chirality affects mostly the axial and the circumferential CTEs, but not the radial CTEs.
1 Introduction
Carbon nanotubes (CNTs) possess high stiffness and
strength and low aspect ratio and density. These extraordinary mechanical properties arouse tremendous interests in CNTs-based nanocomposites [Srivastava &
Atluri (2002), Chung & Namburu (2004), Shen & Atluri
(2004), Nasdala, Ernst & Lengnick (2005), Gao & Gao
(2005)]. Thermal conductance and expansion of the
CNTs are two key properties influencing the mechanical behaviors of the nanocomposites in manufacturing
and operation. Electrically, a single wall carbon nanotube (SWNT) can be either metallic or semi-conducting
depending on its chirality, leading to the possibility to
create CNT-based nanoscale electronic device components [Maiti (2002)]. The observation that conductance
of a metallic CNT changes by orders in magnitude when
strained also opens the door to the potential application
of strain-tuned nanoscale electronic transducer, transistor and switcher [Yang, Han & Anantram (2002)]. The
thermal properties of CNTs also play critical roles in controlling the performance and stability of these nanoscale
electronic components [Liew, Wong & He (2005)].
Many research efforts have been made to determine the
specific heat of CNTs, both theoretically and experimentally. Yi, Lu, Zhang, Pan & Xie (1999) experimentally indicated that over the temperature range of
10 − 300o K, the specific heat of multiwall carbon nanotubes (MWNTs) follows a linear temperature dependence, which they attributed to the constant phonon spectrum. Their results indicated that the out-of-plane acoustic mode (as in a graphene sheet) dominated the heat
capacity. Mizel, Benedict & Cohen (1999) measured
the specific heat for MWNTs in the temperature range
1 < T < 200o K and found a quadratic temperature dependence of the specific heat at low temperature (< 50o K)
and a linear temperature dependence above that. Hone,
Batlogg, Benes & Johnson (2000) and Popov (2002)
made similar observations as Mizel, Benedict & Cohen
(1999). Cao, Yan & Xiao (2003) calculated the specific
heat using a two-atom unit cell model and the lattice dynamics. The specific heat was found to be proportional
to the tubule diameter at low temperatures and inversely
proportional to the square of the diameter at high temperatures. Zhang, Xia & Zhao (2003) used a continuum based model to calculate the phonon dispersion relations for SWNTs, based on which they found that the
axial lattice wave propagations contributed the most to
the specific heat. Li & Chou (2005) calculated the specific heat of SWNTs using molecular structural mechanics and showed that the specific heat increased with increasing tube diameter within the temperature range of
25 − 350o K.
Studies on the thermal expansion of CNTs are very limited. Due to the difficulty in nanoscale experiments,
most of the experiments focused on CNT bundles and
ropes. Ruoff & Lorents (1995) suggested that the radial
coefficient of thermal expansion (CTE) of MWNTs be
1 Center for Aerospace Research and Education, University of Caliessentially identical to the axial CTE. The radial CTE
fornia at Irvine, 5251 California Ave., Suite 140, Irvine, CA 92612
92
c 2006 Tech Science Press
Copyright of MWNTs was found to increase with temperature,
nearly identical to that of the c-axis thermal expansion
of graphite [Bandow (1997)]. The average tube diameter was observed to increase with increasing growth
temperature [Bandow & Asaka (1998)]. Maniwa, Fujiwara, Kira & Tou (2000) reported a radial CTE range of
1.6 × 10−5 − 2.6 × 10−5 /K for the MWNTs. The X-ray
studies by Yosida (2000) and Maniwa, Fujiwara & Kira
(2001) on SWNT bundles suggested negative radial CTE
at low temperatures and positive radial CTE at high temperatures. Although the CTE of SWNT is of fundamental
importance to both the nanoelectronics and nanocomposites, experimental data are not available on the CTE for
individual SWNT. Theoretical investigations of the thermal expansion of SWNTs are also lacking, and sometimes with contradicting results. Raravikar, Keblinski
& Rao (2002) performed MD simulations on (5, 5) and
(10, 10) nanotubes and reported temperature independent
positive values for both the radial and axial CTEs. The
MD simulations by Kwon, Berber & Tománek (2004) indicated negative CTES for SWNTs up to 900o K. Jiang
& Liu (2004) showed that both the radial and axial CTEs
of SWNTs are negative at low temperature but positive at
high temperature, but they did not consider the multibody
interactions in deriving the atom vibrating frequencies.
Lately, the molecular structural approach by Li & Chou
(2005) indicated that both the axial and radial CTEs were
positive and increase with increasing temperature.
In this paper, we endeavor to analyze the specific heats
and the thermal expansion based on a lattice-based cell
model. In Section 2, the framework of the cell model for
calculating the specific heat and the thermal expansion
coefficient is presented. In Section 3, we present results
and analysis of the calculated specific heats and CTEs
versus temperature and tube radii for different tube chiralities. Section 4 summarizes the work.
CMES, vol.14, no.2, pp.91-100, 2006
xA = r, yA = zA = 0, where r is the radius of the tube.
The polar coordinates of atom B are given by (r, ϕB, zB ),
where ϕB = cos−1 xB /r. The positions of atoms C, D are
similarly given. The second nearest neighbor atoms are
located using the nearest neighbor atom coordinates. For
instance, the equivalence of bond BB1 and CA yields
ϕB1 = ϕB + (ϕA − ϕC ) = ϕB − ϕC ,
(1)
zB1 = zB + (zA − zC ) = zB − zC .
(2)
Similarly, the positions of B2, C1, C2, D1, D2 can be
derived.
(a) Tubular SWNT
2 Cell model for SWNT using the local harmonic
approach
(b) Unrolled planar structure
The cell model for SWNT is illustrated in Figure 1. In
Figure 1, the representative atom A is surrounded by
three nearest neighbor atoms B, C and D, forming a lattice cell that can be taken as the basic element of the
tube. The second nearest neighbor atoms B1, B2, C1,
C2, D1, D2 interact with A through multibody atomistic
potentials (e.g., the Tersoff-Brenner potential in below).
Now we introduce a polar coordinate system such that
Figure 1 : Cell model of SWNT in the tubular and unrolled planar structures.
SWNTs can be imagined as a rolled graphene sheet. The
unrolled planar graphene sheet, as illustrated in Figure
1(b), can be visualized by cutting the SWNT along its axial direction followed by “unrolling” it without stretching
93
Cell model for calculating thermal capacity & expansion of SWNT
to the tangent plane at A. In the planar graphene, we set a
2D Cartesian coordinate system such that xA = 0, yA = 0.
Then, the positions of the nearest neighbor atoms in
the 2D Cartesian system are given by (rϕi , zi ), where
i = B,C, D. The graphene basis vectors a1 and a2 are
now given by
−→
a1 = BD = [r(ϕD − ϕB ), zD − zB ],
(3)
vibration coupling among different atoms, thus providing an computationally efficient and conceptually simple
method to calculate the free energy of a system.
In order to calculate the total potential Utot , the interatomic potential is introduced herein. For carbon
atoms, we employ the Tersoff-Brenner potential [Brenner (1990)], which is expressed as
Vi j = VR(ri j ) − B i jVA (ri j ),
(9)
−→
a2 = CD = [r(ϕD − ϕC ), zD − zC ] .
(4) for atoms i and j, where ri j is the distance between them.
The repulsive and attractive terms are given by
In carbon nanotubes (CNTs), the graphene is rolled up in
such a way that a graphene lattice vector c = na1 + ma2
D(e) −√2Sβ(r−R(e) )
fc(r),
(10)
becomes the circumstance of the tube, where the chirality VR (r) = S − 1 e
(n, m) uniquely determines the tube. Utilizing the equations (3) and (4), the circumstance of the tube is now
D(e) S −√2/Sβ(r−R(e))
given by
e
fc (r),
(11)
VA (r) =
S−1
√
|c| = c · c
where the function fc is a smooth function used to limit
r2 [n(ϕD − ϕB ) + m(ϕD − ϕC )]2
=
the range of the potential to
neighbor atoms,
the nearest
1
(1)
i.e., f c (r) = 1, 2 {1 + cos π(r − R )/(R(2) − R(1)) }, 0
2
+ [n(zD − zB ) + m(zD − zC ) ] .
(5)
for r < R(1), R(1) < r < R(2), r > R(2) , respectively. The
parameter B i j takes account of the multibody interaction
Meanwhile,
through the bond angles formed at atom i, and is given
|c| = 2πr.
(6) by
1
2
Equations (5) and (6) yield
g = r2 [n(ϕD − ϕB ) + m(ϕD − ϕC )]2 − 4π2
+ [n(zD − zB ) + m(zD − zC ) ]2 ≡ 0,
B i j = (Bi j + B ji ) ,
(7)
where
(12)
where g is a geometric constraint that connects the tube Bi j = 1 + ∑ G θi jk fc (rik
k(=i, j)
chirality to the coordinate variables.
LeSar, Najafabadi & Srolovitz (1989) proposed the local
harmonic (LH) approach to calculate the Helmholtz free
energy for a finite-temperature equilibrium atomic solid.
At the heart of the LH approach is the local description
of the atomic vibrations, i.e.,
2
2
1
∂
U
tot
= 0, i = 1, 2, . . . N,
ω I3×3 −
(8)
iκ
mC ∂xi ∂xi where mC is the carbon atom mass, Utot is the total potential energy of the system, and ωi is the vibrating frequencies of atom i (varying from 1 to N – the total number of
atoms) in the κ(= 1, 2, 3) direction determined with the
rest atoms fixed at their equilibrium positions (as implied
by the partial derivatives). The LH model neglects the
−δ
,
c20
c2
,
G (θ) = a0 1 + 02 − 2
d0 d0 + (1 + cos θ)2
(13)
(14)
2
− r2jk /2ri j rik defines the anand where cos θi jk = ri2j + rik
gle subtended by the adjoint carbon bonds i − j and i − k.
The material parameters employed in this paper are given
in the Appendix.
The Helmholtz free energy using the local harmonic
model is now given as [LeSar, Najafabadi & Srolovitz
(1989), Foiles (1994)]:
N 3
hωiκ
,
(15)
H = Utot + kB T ∑ ∑ ln 2 sinh
4kB T
i=1 κ=1
94
c 2006 Tech Science Press
Copyright CMES, vol.14, no.2, pp.91-100, 2006
where kB and h are the Boltzmann and the Planck’s constants, respectively. The total potential Utot directly influenced by a change of atom A’s position is reflected in
those bonds of the first closest layers, i.e., AB, AC, AD,
and through the bond angles in those of the second closet
layers, i.e., BB1, BB1,CC1,CC2, DD1, DD2. Therefore,
the total energy can be expressed as
Utot = VAB +VAC +VAD
+VBB1 +VBB2 +VCC1 +VCC2 +VDD1 +VDD2
where
1
Ua = (VAB +VAC +VAD )
(20)
2
is the potential energy per atom.
Therefore, the
Helmholtz free energy per atom Ha for the LH approach
can be expressed as
3
hωAκ
Ha = Ua + kB T ∑ ln 2 sinh
.
(21)
4kb T
κ=1
The equilibrium atom positions for SWNTs at homoge(16) nous finite temperature T can be obtained by solving for
the minimum of Ha , i.e.:
Jiang & Liu (2004) erroneously neglected the multibody
energy contributions represented by the second line on ∂Ha = 0, ∂Ha = 0, and ∂Ha = 0,
(22)
∂zi
∂r
the right-hand side of equation (16), as the second nearest ∂ϕi
neighbor atoms (B1, B2, C1, C2, D1, D2) could not be where i = B,C, D. Note that the minimization is subcharacterized in their model.
jected to the nonlinear chirality constraint g ≡ 0.
The vibrating frequencies at atom A can now be derived Once the equilibrium configuration is solved, the specific
by substituting the total energy Utot (16) into equation heat per mass is given by [Jiang & Huang (2005)]
(8). In doing so, we point out the seven primary variables
d ln ωAκ
1
3
−
ω2Aκ
T
dT
kB
(namely, unknowns) in our model, i.e., (ϕB , zB ), (ϕC , zC ),
(23)
Cv =
∑ sinh2 (ωAκ ) ,
(ϕD , zD) and r. The second nearest atoms are specified
mC κ=1
using the bond equivalence as represented by equations
where
(1) and (2). It can be readily proved for a SWNT of a
homogenous temperature T , the frequencies ωiκ are in- ωAκ = hωAκ ,
(24)
4πkB T
dependent of the atom i and can be taken the same as
those of the representative atom A. However, we need is the dimensionless frequency. For numerical conveto mention that although the global diagonalization from nience, (23) is approximated using a backward difference
the quasiharmonic to the local harmonic approaches de- scheme.
couples the vibrating among atoms, the directional cou- The thermal expansion is characterized by the coefficient
pling within the local harmonic model cannot be readily of thermal expansion, defined as
taken as null. Hence, the off-diagonal component in the
1 dli
,
(25)
local dynamic matrix cannot be neglected as did in Jiang αi =
li dT
& Huang (2005).
where li is the instantaneous length given by
Now the Helmholtz free energy of the system at finite
temperature can be obtained as
lz = max (zB − zC , zD − zC ),
3
hωAκ
,
(17) lr = r,
H = Utot + kB T N ∑ ln 2 sinh
4kB T
κ=1
+ bond energies independent of atom A.
where ωAκ is also a function of the primary variables.
The total potential energy Utot for the Tersoff-Brenner
formalism can be written as
1
(18)
Utot = ∑ ∑ Vi j .
2 i j=i
Using the bond equivalence, it can be shown that
Utot = NUa ,
(19)
lc = r max (ϕD − ϕB , ϕD − ϕC ) ,
where z, r, c represent respectively the axial, radial and
circumferential directions. In our numerical implementations, a central finite difference scheme is employed to
approximate (25), i.e.,
αi =
1 liT +ΔT − liT −ΔT
.
2ΔT
liT
(26)
95
Cell model for calculating thermal capacity & expansion of SWNT
3 Results and discussions
1500
Cv (mJ/g − K)
Figure 2 shows the calculated Cv for the armchair (n, n)
SWNTs. In Figure 2(a), for comparison purpose, the specific heats for graphite and diamond are also shown. In
Figure 2(b), the calculated Cv ’s are shown for the low
temperature range of 2 − 300o K, together with Hone et
al. [Hone, Batlogg, Benes, et al. (2000)] measured data
for SWNT ropes. Over a wide range of temperature simulated, especially for high temperature above 100o K, the
present analysis agrees well with the experimental data.
The results indicate that the specific heats do not depend
on the radius of the nanotubes, except at temperature
range of 2 − 300oK. This high temperature independence
of the specific heat was attributed to the phonon states of
the constituent graphene sheet [Hone, Laguno & Biercuk
(2002)]. The calculated high temperature specific heats
approach the theoretical limit value of 2078mJ/g − K,
regardless of the the chirality and radius of the tube.
1000
Charility
(5,5)
(10, 10)
(20, 20)
500
Exp. data for graphite
[32]
Exp. data for diamond
[33]
[34]
Exp. data for diamond
0
0
500
T (o K)
1000
(a) 2 − 1000K
Cv (mJ/g − K)
600
Figure 3 shows the specific heats versus the radius of
Charility
(5,5)
the nanotubes at three levels of temperature. The open
(10, 10)
500
symbols represent the data for the armchair tubes, while
(20, 20)
the the filled tubes represent those for the zigzag tubes
(25, 25)
400
(5, 0), (10, 0), (20, 0), (30, 0). Figure 3 further conExp. data [12]
firms that the specific heats are radius independent at
300
high temperature, and that at low temperature, the specific heats show a strong dependence on the tube ra200
dius for small tubes and a weak dependence on the tube
radius for large tubes. No obvious differences are ob100
served for the armchair and zigzag tubes. We also calculated the Cv for different tube chiralities (25, m), where
0
m = 0, 3, 6, 9, 15, 20,25. Our results (not shown here) in0
100
200
300
dicate that Cv is only very slightly dependent on the chiT (o K)
ralities of the tubes and that Cv ’s for the chirality tubes
(b) 2 − 300K
(with m = 3, . . ., 20) are contained in those of the armchair and the zigzag tubes.
Figure 2 : Temperature dependence of specific heat for
Our results deviates below the experimental data for tem- (n, n) SWNTs.
perature lower than 100oK. Figure 4 shows the contributions to the specific heat from the three atom vibrating
modes for (10, 10) tube. As observed by Yi, Lu & Zhang
(1999), at low temperature, our results clearly show that (expect κ = r) is quite independent on the radius of the
the out-of-plane vibrating mode (the radial mode) domi- tubes. In Figure 5, we also plot the horizontal line of
nates the specific heat of the SWNT. As the temperature ln(2 sinhω) = 0, namely, ω = 0.48. Above about 500o K,
increases, the contributions from the circumferential and ωr turns to increase, instead of decreasing, the Helmholtz
then the axial vibrating modes gradually increase. Fig- free energy. The radial mode is unique to the SWNTs
ure 5 gives the normalized atom vibrating frequencies [Ravavikar, Keblinski & Rao (2002)]. The discrepancy
ωκ for the armchair tubes (n, n). It can be seen that ωκ of the calculated Cv at low temperature is caused by the
c 2006 Tech Science Press
Copyright 96
CMES, vol.14, no.2, pp.91-100, 2006
2000
1400
1200
1000
Total
800
Axial
Circumferential
1500
Cv (mJ/g − K)
Cv (mJ/g − K)
600
T
400
100
300
200
500
Radial
1000
500
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0
1.8
0
200
400
600
800
1000
1200
1400
T (o K)
r(nm)
Figure 3 : Radius dependence of specific heats for Figure 4 : Mode contributions to specific heats for
(10, 10) tube.
SWNTs.
overestimated radial vibrating frequency, which should
be much more constrained by the compressed out-ofplane π bonding orbitals. In the original formalism of
Brenner’s potential [Brenner (2000)], the π bond is reflected in the multibody term B i j as
Chirality
(5,5)
(10,10)
(20,20)
101
(27)
where
DH
Biπj = ΠRC
i j + Bi j ,
2
ωκ
1
2
B i j = (Bi j + B ji ) + Biπj ,
10
ωz
ωc
(28)
and where the first term ΠRC
i j represents the influence of
radical energetics and π bond conjugation on the bond
energies, and the second term BiDH
j depends on the dihedral angle for carbon-carbon double bonds. We tried
a constant Biπj = −0.0243 (taken from Brenner [Brenner (1990)] for grahpite) in our calculations. The results
overpredicted the low temperature specific heat (comparing to the experimental data) by nearly two times, but
approached the experimental values at high temperature
(starting at ∼ 300o K). Due to lack of Biπj data and their
derivatives for SWNTs, we made no further attempt in
enhancing the calculated low temperature specific heat.
Figure 6 shows the calculated CTE for the armchair,
zigzag and chirality tubes. In each case, the axial and
circumferential CTEs are also zeroes at 2o K, but being
shifted an amount upwards for clarity. In accordance
10
10
0
ωr
-1
0
500
1000
1500
T (o K)
Figure 5 : Normalized atom vibrating frequencies for the
armchair tubes.
with Li et al.’s [Li & Chou (2005)] calculations, our results indicate universal positive CTEs for the radial, axial and circumferential directions. As can be seen from
6(a), for the armchair (n, n) tubes, the smaller tubes have
slightly higher axial but lower radial CTEs than the larger
tubes. The two large tubes (20, 20) and (25, 25) shows
97
Cell model for calculating thermal capacity & expansion of SWNT
1E-05
CTE
8E-06
αc
6E-06
4E-06
Chirality
αz
(5,5)
(10,10)
2E-06
(20,20)
αr
0
0
200
400
(25, 25)
600
800
T (o K)
1000
1200
1400
1000
1200
1400
(a) (n, n)
Chirality
(5,0)
1.6E-05
(10,0)
(20,0)
(30,0)
1.2E-05
CTE
The CTE variations for the zigzag tubes are shown in Figure 6(b). For the zigzag tubes, the tube radius has a more
pronounced effect on the thermal expansion. The large
tube (30, 0) has a radial CTE that is nearly two to three
times larger than that of the small tube (5, 0). But the axial CTE for (30, 0) is about two to three times lower than
that for the small tube (5, 0). Very high axial CTEs are
observed for the small zigzag tube. Note that the axial
direction for the zigzag tubes is the circumferential direction for the armchair tubes. Hence, this observation
of high axial CTEs for the zigzag tubes is in accordance
with the above observation of high circumferential CTEs
for the armchair tube. The explanation for this preference of thermal expansion is simple. Take the armchair
tube (10, 10) for instance. Table 1 gives the atom positions in the 2D Cartesian system for T = 300, 1000oK.
It is seen that atom B moves slightly away from atom A
(at xA = yA = 0) nearly along the circumference, and that
C moves slightly away from A nearly along the axial direction. But atom D is also seen to move slightly away
along the circumference. Bond AD’s circumferential extension, together with bond AB circumferential extension
and counterclockwise rotations, causes the large circumferential CTE for the armchair tubes. For the zigzag
tubes, no paradox is seen for the circumferential CTEs,
which now follows nearly identical paths as the radial
CTEs.
1.2E-05
8E-06
αz
4E-06
0
αc
αr
0
200
400
600
800
T (o K)
(b) (n, 0)
1E-05
8E-06
CTE
almost the same CTEs, which means that the CTEs are
radius independent for large tubes. To the best the author’s knowledge, the circumferential CTE is for the first
time reported in the literature. Interestingly, for the armchair tubes, the radial and the circumferential CTEs do
not follow the same path. For instance, the small tube
(5, 5) has a lower αr but a much higher αc than those
of (10, 10), although one might expect αc = αr . This
seemingly contradiction reflects the self-adjusting of the
unit cell composed of atoms A, B, C, D in the minimization process of the Helmholtz free energy. For the small
tubes, the paradox suggests more self-extension in the
circumference.
6E-06
αz
4E-06
αc
Chirality
(25,0)
2E-06
(25,9)
αr
0
0
(25,25)
200
400
600
800
1000
1200
1400
Figure 6(c) shows the effect of the chirality on the therT (o K)
mal expansion. Opposite temperature effects are seen for
(c) (25, m)
the axial and the radial CTEs. Comparing the armchair
and the zigzag tubes, the zigzag tube (25, 0) has a much Figure 6 : Temperature and chirality dependence of
larger axial CTE and a relatively lower radial CTE. And CTEs for SWNTs.
as seen above, the armchair tube has a high circumferential CTE. We need to mention that small local oscillations
98
c 2006 Tech Science Press
Copyright CMES, vol.14, no.2, pp.91-100, 2006
Table 1 : Equilibrium atom positions for (10, 10). The length unit is 10−2nm
.
300
1000
B
(-7.294, 12.597)
(-7.330, 12.600)
C
(-7.267, -12.635)
(-7.264, -12.688)
D
(14.603, 0)
(14.640, 0)
are seen on the calculated CTEs. This small local oscilla- Appendix
tions are introduced by the built-in convergence criteria
The parameters for Brenner’s potentials [Brenner
of the IMSL optimizaiton package we used. Small and
(1990)]:
artificial, they are smoothed out by a polynomial fitting.
D(e) = 6.0eV,
R(1) = 0.17nm,
S = 1.22,
β = 21nm−1 ,
R(2) = 0.27nm,
a0 = 0.00020813,
c0 = 330,
R(e) = 0.139nm,
d0 = 3.5.
4 Summary
In summary, we develop a lattice-based cell model to
calculate the specific heats and the coefficients of thermal expansion of single wall carbon nanotubes. The
cell model consists of seven primary coordinate variables
while subjecting to a chirality constraint.
The specific heat and thermal expansion of the SWNTs
are studied. The calculated specific heats are in good
agreement with experimental data and for a large range
of temperatures, very close to those of graphite and diamond. The chirality dependence of the specific heat is
seen only up to a few hundred Kelvins. Small tubes have
much lower values at low temperature. Positive thermal
expansions are observed for all the radial, axial and circumferential directions. The coefficients of thermal expansion (CTEs) increase with increasing temperatures.
For the armchair tubes, both the radial and axial CTEs
are only slightly influenced by the tube radii. But the
armchair tubes are seen to have large values for the circumferential CTEs. The zigzag tubes have small radial
and circumferential CTEs, which also weakly depend on
the tube radii. But the zigzag tubes see very large axial
CTEs, which also strongly depend on the tube radii. The
axial and circumferential CTEs, but not the radial ones,
are much influenced by the tube chiralities.
Acknowledgement: This work is supported by the
Army Research Laboratory. The first author would also
like to thank Dr. Brenner for providing his nanotube code
that generates the zero temperature (n, m) carbon nanotubes.
The carbon bond length for the zero temperature
graphene sheet is set to 0.14507nm. The other constants
are given as: mc = 1.9926 × 10−26Kg. The Planck’s constant h = 6.626068 × 10−34Js. And the Boltzmann constant kB = 1.3806503 × 10−23JK −1 .
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